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Inspired by consideration in heterotic string theory, we construct a morphismof spectra from $\mathrm{KO}((q))/\mathrm{TMF}$ to $\Sigma^{-20}I_{\mathbb{Z}}\mathrm{TMF}$, which we show to be an isomorphism and to implement the Andersonself-duality of $\mathrm{TMF}$. We will use this to factor a related morphismfrom $\mathrm{TMF}$ to$\Sigma^{-20}I_{\mathbb{Z}}(\mathrm{MSpin}/\mathrm{MString})$. This lattermorphism induces a differential geometric pairing, which captures not only theinvariant of Bunke and Naumann but also a finer invariant which detects subtleAnderson dual pairs of elements of $\pi_\bullet \mathrm{TMF}$. Our analysisleads to a conjecture concerning certain self-dual vertex operatorsuperalgebras and some specific torsion classes in $\pi_\bullet \mathrm{TMF}$,which we will detail in an appendix.  This paper is written in the rigorous mathematical style, except in a sectionwhich summarizes and translates the content into a language more amenable tostring theorists. In physics terms, our result allows us to compute thediscrete part of the Green-Schwarz coupling of the $B$-field in a couple ofsubtle hitherto-unexplored cases.

Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras